Enter your lower bound (a) and upper bound (b) to compute key uniform distribution statistics. Specify an interval [x₁, x₂] to get the probability P(x₁ ≤ X ≤ x₂), plus the PDF, CDF, mean, median, variance, and standard deviation — all calculated for your continuous uniform distribution U(a, b).
P(x₁ ≤ X ≤ x₂)
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Probability Density Function f(x)
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CDF at x₁ — F(x₁)
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CDF at x₂ — F(x₂)
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Mean (μ)
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Median
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Variance (σ²)
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Standard Deviation (σ)
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The uniform distribution is a continuous probability distribution where all outcomes within a given interval [a, b] are equally likely. The probability is spread evenly across the interval, meaning every sub-interval of equal length has the same probability. It is sometimes called the rectangular distribution because its PDF graph looks like a rectangle.
For a continuous uniform distribution U(a, b), the probability that X falls between x₁ and x₂ is P(x₁ ≤ X ≤ x₂) = (x₂ − x₁) / (b − a), provided a ≤ x₁ < x₂ ≤ b. Simply subtract the lower interval bound from the upper, then divide by the total range of the distribution.
The mean (expected value) of a continuous uniform distribution U(a, b) is μ = (a + b) / 2. It is simply the midpoint of the interval [a, b]. For example, if a = 0 and b = 10, the mean is 5.
The median of a continuous uniform distribution equals the mean, which is (a + b) / 2. Because the distribution is symmetric over [a, b], the median and the mean coincide at the midpoint of the interval.
The variance of U(a, b) is σ² = (b − a)² / 12, and the standard deviation is σ = (b − a) / √12. For instance, with a = 0 and b = 10, the standard deviation is 10 / √12 ≈ 2.887.
The PDF of a continuous uniform distribution is f(x) = 1 / (b − a) for a ≤ x ≤ b, and 0 otherwise. This constant value reflects the fact that every point in [a, b] is equally likely. The PDF is flat (constant), which is why the shape is rectangular.
The CDF F(x) gives the probability that X is less than or equal to a specific value x. For U(a, b): F(x) = 0 for x < a, F(x) = (x − a) / (b − a) for a ≤ x ≤ b, and F(x) = 1 for x > b. It increases linearly from 0 to 1 over the interval [a, b].
No, they are different. The normal distribution is bell-shaped, with most values clustering around the mean and probabilities tapering off toward the tails. The uniform distribution assigns equal probability to all values in [a, b] — its PDF is flat, not bell-shaped. The two distributions share no functional form.